Pierre de Fermat

(1601–65; b. Beaumont-de-Lomagne, France; d. Castres, France) French mathematician. He is particularly famous for his 'last theorem', which he discovered in about 1637, and of which he claimed he had a 'marvellous demonstration'. He died without revealing his proof and it was not until 1994 that the English mathematician Andrew Wiles gave a full proof. Fermat's father was a wealthy leather merchant. Fermat was educated at a Franciscan monastery and studied law at U Toulouse. He became a judge, but he had a passion for mathematics and obtained many mathematical theorems, which he communicated to fellow mathematicians, always remaining very secretive about his proofs. The correspondence between Fermat and Pascal laid the foundations of the modern theory of probability. A lunar crater and a street in Paris are named after him.

(click to enlarge)
Fermat, portrait by Roland Lefèvre; in the Narbonne City Museums, France (credit: Courtesy of the Musée de la Ville de Narbonne, France)

(born Aug. 17, 1601, Beaumont-de-Lomagne, France — died Jan. 12, 1665, Castres) French mathematician. Of Basque origin, Fermat studied law at Toulouse and developed interests in foreign languages, Classical literature, ancient science, and mathematics. A jurist by profession, he produced major mathematical breakthroughs independently and collaboratively. A contemporary of René Descartes, he discovered independently the basic principles of analytic geometry, but, because Fermat's work was published posthumously, the field became known as Cartesian geometry. He found equations for tangent lines to curves through processes equivalent to differentiation and was coauthor (with Blaise Pascal) of probability theory. His work in number theory, especially divisibility, led to some of its most important theorems. He seldom demonstrated his results, which led to a centuries-long quest to prove a famous conjecture that Fermat claimed was easily shown (see Fermat's last theorem).

For more information on Pierre de Fermat, visit Britannica.com.

[b. Beaumont-de-Lomagne, France, August 17, 1601, d. Castres, near Toulouse, France, January 12, 1665]

Fermat was a lawyer whose thoughts on mathematics were transmitted mainly by correspondence with other mathematicians. He discovered analytic geometry a year before Descartes had the same idea. Fermat employed some of the main ideas of calculus before either Newton or Leibniz. His main fame today, however, comes from his work in number theory. Fermat's "last theorem," which is that for natural numbers x, y, and z there is no natural number n greater than 2 for which xⁿ+ yⁿ= zⁿ is true, is especially well known. Fermat claimed a proof of this theorem, but did not reveal it. It was finally proved in 1995.

The French mathematician Pierre de Fermat (1601-1665) played an important part in the foundation and development of analytic geometry, the calculus of probabilities, and especially the theory of numbers.

Pierre de Fermat was born on Aug. 17, 1601, at Beaumont-de-Lamagne near Montaubon. There is some doubt as to the precise date of his birth. He is said to have been baptized on Aug. 20, 1601, but his tombstone puts his birth as 1608, and others have stated 1595. He was the son of Dominique Fermat, a leather merchant, and Claire de Long.

It was decided that Fermat should be trained as a magistrate, and he was sent to Toulouse. The general lines on which he was educated can only be guessed, and so far as his career as jurist is concerned, there is a record of his installation at Toulouse on May 14, 1631. In 1648 he was promoted to king's counselor in the Parliament of Toulouse, a post which he held until his death on Jan. 12, 1665. In 1631 he married his mother's cousin, Louise de Long; they had three sons and two daughters.

Theory of Numbers

It was perhaps C. G. Bachet's translation (1621) of Diophantus of Alexandria that stimulated Fermat's interest in the theory of numbers. That same edition was republished in 1670, with the addition of Fermat's notes edited by his son, who tells of the immense difficulties of collecting his father's writings, because they were only known from letters and notes in which Fermat usually stated theorems without proof. Much of Fermat's notes on Diophantus's problems were taken from the margin of his copy of Bachet's work.

As a specimen of Fermat's genius, there is his theorem that every number is either a square or the sum of two, three, or four squares. He arrived at his proof after long attempts to break down the solution into a multitude of minor solutions. In so doing, he also found many lesser but still important results. His final technique made use of his method of infinite descent, which may be faintly appreciated from a quotation. After saying that the theorem alluded to was beyond the power of René Descartes, by his own admission, Fermat goes on: "I have at last brought this under my method, and I prove that, if a given number were not of this nature, there would exist a number smaller than it which would not be so either, and again a third number smaller than the second, etc. ad infinitum; whence we infer that all numbers are of the nature indicated." Later, to prove an even more general proposition, he had to prove first five lesser theorems, and there again he made use of the same technique. The first of these theorems is worth singling out for comment. It is the theorem that every prime number of the form 4n + 1 is the sum of two squares (thus 4 x 1 + 1 = 1² + 2²;4x2 + 1 = 0² + 3²;4x3 + 1 = 2² + 3²; and so on). The great mathematician Leonhard Euler was the first to supply and publish a proof of this (1770).

There is one theorem, however, which has never been proved, and this has become something of a legend as Fermat's "Last Theorem," which states that there is no solution in integers of the equation xⁿ + yⁿ = zⁿ (xyz ≠ 0, n >2). There is a method in Diophantus for dividing a given square into two squares. Against this proposition in the margin of Bachet's edition Fermat wrote the following note: "On the other hand, it is impossible to separate a cube into two cubes, a fourth power into two fourth powers, or, generally, any power except a square into two powers with the same exponent (i.e., of the same degree). I have discovered a truly marvelous proof of this, which however, the margin is not large enough to contain." Did Fermat have a proof of what, for 3 centuries, others have failed to prove? Euler proved it when the exponent (n) was equal to 3 or 4. Others have extended enormously the range of related theorems which are provable, but none has found a "truly marvelous proof" such as Fermat claimed to have found.

Analytical Geometry

Some of Fermat's first original mathematics appears to have been inspired by a famous problem of Apollonius. The crucial problem was this: if from a point in a plane four fixed lines are drawn, and four other lines through a point P cross the first four, all at the given angle; and if, furthermore, the distance along the four variable lines from P to where they cross the others is known as a, b, c, and d; then if a · c is in constant ratio to b · d, the point P moves on a conic section (an ellipse, parabola, hyperbola, circle, or pair of straight lines). This theorem, which is not easy to write more concisely in simple language, may be called the "four-line theorem." If two of the fixed lines coincide, then the path of P is still a conic, but the problem, the "three-line problem," is easier.

Fermat provided his own proof of the three-line theorem, and in doing so he made use of analytical methods, determining points in a plane by two coordinates, showing that if the coordinates are related by equations like 2x + 3y = 5, then the point lies on a straight line, and so on. He also worked out the equations of the curves known as conic sections, and he was quite familiar with coordinate methods in three dimensions.

Other work of a similar character by Fermat relates to the problem of constructing a tangent to a curve using infinitesimals. He found a method of calculating the length of a curve (involving the method of tangents) by first solving a problem of areas. Almost as important was his method of maxima and minima. This was first published in 1638 and was used for finding centers of gravity.

Principle of Light Transmission

It was shown by Hero of Alexandria that light traveling between two points, and undergoing a reflection in the process, follows the shortest path. For example, to follow the shortest path, light passing through a bowl of water would travel in a straight line; but observation shows that this is not true. Fermat demonstrated that such refracted (bent) light must be measured by optical distance, and this is always a minimum. The optical distance is the sum of the products of the distances and the corresponding refractive indexes. Fermat's conception involved the (correct) belief that light travels more slowly in more optically dense media. As subsequently developed, it was of great importance to the derivation of geometrical optics and also influenced J. Bernoulli in founding the calculus of variations.

Fermat did important work in the foundation of a theory of probability, which grew out of his early researches into the theory of numbers. In this theory he was without equal in his century and perhaps in any century. By any standards he was a great mathematician, but his name is less often encountered than it would have been had his personal life been better known.

Further Reading

L. J. Mordell, Three Lectures on Fermat's Last Theorem (1921), gives a short résumé of Fermat's life. Eric Temple Bell, Men of Mathematics (1937), contains a chapter on Fermat. See also James R. Newman, ed., The World of Mathematics (4 vols., 1956), and Joseph Frederick Scott, A History of Mathematics: From Antiquity to the Beginning of the Nineteenth Century (1958).

Pierre de Fermat
Pierre de Fermat
Born	Beaumont-de-Lomagne, France
Died	12 January 1665 Castres, France
Residence	France
Nationality	French
Fields	Mathematics and Law
Known for	Analytic geometry Probability Fermat's Last Theorem

Pierre de Fermat (French pronunciation: [pjɛːʁ dəfɛʁˈma]; 17 August 1601 or 1607/8^[1] – 12 January 1665) was a French lawyer at the Parlement of Toulouse, France, and an amateur mathematician who is given credit for early developments that led to modern calculus. In particular, he is recognized for his discovery of an original method of finding the greatest and the smallest ordinates of curved lines, which is analogous to that of the then unknown differential calculus, as well as his research into the theory of numbers. He also made notable contributions to analytic geometry, probability, and optics. He is best known for Fermat's Last Theorem, which he described in a note at the margin of a copy of Diophantus' Arithmetica.

Contents 1 Life and work 1.1 Work 1.2 Death 2 Assessment of his work 3 See also 4 Notes 5 References 6 Further reading 7 External links

Life and work

Fermat was born at Beaumont-de-Lomagne, Tarn-et-Garonne, France. The late 15th century mansion where Fermat was born in Beaumont-de-Lomagne is now a museum. Pierre Fermat's father was a wealthy leather merchant and second consul of Beaumont-de-Lomagne. Pierre had a brother and two sisters and was almost certainly brought up in the town of his birth. There is little evidence concerning his school education, but it may have been at the local Franciscan monastery.

He attended the University of Toulouse before moving to Bordeaux in the second half of the 1620s. In Bordeaux he began his first serious mathematical researches and in 1629 he gave a copy of his restoration of Apollonius's De Locis Planis to one of the mathematicians there. Certainly in Bordeaux he was in contact with Beaugrand and during this time he produced important work on maxima and minima which he gave to Étienne d'Espagnet who clearly shared mathematical interests with Fermat. There he became much influenced by the work of Franciscus Vieta.

From Bordeaux Fermat went to Orléans where he studied law at the University. He received a degree in civil law before, in 1631, receiving the title of councillor at the High Court of Judicature in Toulouse, which he held for the rest of his life. Due to the office he now held he became entitled to change his name from Pierre Fermat to Pierre de Fermat. Fluent in Latin, Greek, Italian, and Spanish, Fermat was praised for his written verse in several languages, and his advice was eagerly sought regarding the emendation of Greek texts.

He communicated most of his work in letters to friends, often with little or no proof of his theorems. This allowed him to preserve his status as an "amateur" while gaining the recognition he desired. This naturally led to priority disputes with fellow contemporaries such as Descartes and Wallis. He developed a close relationship with Pascal.

Anders Hald writes that, "The basis of Fermat's mathematics was the classical Greek treatises combined with Vieta's new algebraic methods."^[2]

Work

Fermat's pioneering work in analytic geometry was circulated in manuscript form in 1636, predating the publication of Descartes' famous La géométrie. This manuscript was published posthumously in 1679 in "Varia opera mathematica", as Ad Locos Planos et Solidos Isagoge, ("Introduction to Plane and Solid Loci").^[3]

In Methodus ad disquirendam maximam et minima and in De tangentibus linearum curvarum, Fermat developed a method for determining maxima, minima, and tangents to various curves that was equivalent to differentiation.^[4] In these works, Fermat also obtained a technique for finding the centers of gravity of various plane and solid figures, which led to his further work in quadrature.

Pierre de Fermat

Fermat was the first person known to have evaluated the integral of general power functions. Using an ingenious trick, he was able to reduce this evaluation to the sum of geometric series.^[5] The resulting formula was helpful to Newton, and then Leibniz, when they independently developed the fundamental theorem of calculus.^{[citation needed]}

In number theory, Fermat studied Pell's equation, Fermat numbers, perfect, and amicable numbers. It was while researching perfect numbers that he discovered the little theorem. He also invented a factorization method which has been named for him as well as the proof technique of infinite descent, which he used to prove Fermat's Last Theorem for the case n = 4. Fermat also developed the two-square theorem, and the polygonal number theorem, which states that each number is a sum of three triangular numbers, four square numbers, five pentagonal numbers, and so on. Although Fermat claimed to have proved all his arithmetic theorems, few records of his proofs have survived. Many mathematicians, including Gauss, doubted several of his claims, especially given the difficulty of some of the problems and the limited mathematical tools available to Fermat. His famous Last Theorem was first discovered by his son in the margin on his father's copy of an edition of Diophantus, and included the statement that the margin was too small to include the proof. He had not bothered to inform even Mersenne of it. It was not proved until 1994, using techniques unavailable to Fermat.

Although he carefully studied, and drew inspiration from Diophantus, Fermat began a different tradition. Diophantus was content to find a single solution to his equations, even if it were an undesired fractional one. Fermat was interested only in integer solutions to his Diophantine equations, and he looked for all possible general solutions. He also often proved that certain equations had no solution, which usually baffled his contemporaries.

Through his correspondence with Blaise Pascal in 1654, Fermat and Pascal helped lay the fundamental groundwork for the theory of probability. From this brief but productive collaboration on the problem of points, they are now regarded as joint founders of probability theory.^[6] Fermat is also credited with carrying out the first ever rigorous probability calculation. In it, he was asked by a professional gambler why if he bet on rolling at least one six in four throws of a die he won in the long term, whereas betting on throwing at least one double-six in twenty-four throws of two dice resulted in him losing. Fermat subsequently proved why this was the case mathematically.^[7]

Fermat's principle of least time (which he used to derive Snell's law in 1657) was the first variational principle^[8] enunciated in physics since Hero of Alexandria described a principle of least distance in the first century CE. In this way, Fermat is recognized as a key figure in the historical development of the fundamental principle of least action in physics. The term Fermat functional was named in recognition of this role.^[9]

Death

He died at Castres, Tarn, age 63. The oldest, and most prestigious, high school in Toulouse is named after him: the Lycée Pierre de Fermat. French sculptor Théophile Barrau made a marble statue named Hommage à Pierre Fermat as tribute to Fermat, now at the Capitole of Toulouse.

Assessment of his work

Holographic will handwritten by Fermat on 4 March 1660 — kept at the Departmental Archives of Haute-Garonne, in Toulouse

Together with René Descartes, Fermat was one of the two leading mathematicians of the first half of the 17th century. Independently of Descartes, he discovered the fundamental principles of analytic geometry. With Blaise Pascal, he was a founder of the theory of probability.

Regarding Fermat's work in analysis, Isaac Newton wrote that his own early ideas about calculus came directly from "Fermat's way of drawing tangents."^[10]

Of Fermat's number theoretic work, the great 20th century mathematician André Weil wrote that "... what we possess of his methods for dealing with curves of genus 1 is remarkably coherent; it is still the foundation for the modern theory of such curves. It naturally falls into two parts; the first one ... may conveniently be termed a method of ascent, in contrast with the descent which is rightly regarded as Fermat's own."^[11] Regarding Fermat's use of ascent, Weil continued "The novelty consisted in the vastly extended use which Fermat made of it, giving him at least a partial equivalent of what we would obtain by the systematic use of the group theoretical properties of the rational points on a standard cubic."^[12] With his gift for number relations and his ability to find proofs for many of his theorems, Fermat essentially created the modern theory of numbers.

See also

• Fermat cubic
• Fermat's factorization method
• Fermat hypersurface
• Fermat's Last Theorem
• Fermat number
• Fermat's principle
• Fermat Prize of mathematical research
• Fermat pseudoprime
• Fermat's spiral
• Fermat's Theorem (stationary points)
• Fermat-Euler theorem
• Pell-Fermat's Diophantine equation
• List of amateur mathematicians
• Fermat's Room

Notes

1. ^ Klaus Barner (2001): How old did Fermat become? Internationale Zeitschrift für Geschichte und Ethik der Naturwissenschaften, Technik und Medizin. ISSN 0036-6978. Vol 9, No 4, pp. 209-228.
2. ^ http://www.ams.org/notices/199507/faltings.pdf
3. ^ p548, Jan Gullberg, Mathematics from the birth of numbers, W. W. Norton & Company; ISBN 039304002X ISBN 978-0393040029
4. ^ Pellegrino, Dana. "Pierre de Fermat". http://www.math.rutgers.edu/~cherlin/History/Papers2000/pellegrino.html. Retrieved 2008-02-24. 
5. ^ Paradís, Jaume; Pla, Josep; Viader, Pelagrí, Fermat’s Treatise On Quadrature: A New Reading, http://papers.ssrn.com/sol3/Delivery.cfm/SSRN_ID848544_code386779.pdf?abstractid=848544&mirid=5, retrieved 2008-02-24 
6. ^ O'Connor, J. J.; Robertson, E. F., The MacTutor History of Mathematics archive: Pierre de Fermat, http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Fermat.html, retrieved 2008-02-24 
7. ^ Howard Eves, An Introduction to the History of Mathematics, Saunders College Publishing, Fort Worth, TX, 1990.
8. ^ "Fermat’s principle for light rays". http://relativity.livingreviews.org/open?pubNo=lrr-2004-9&page=articlesu9.html. Retrieved 2008-02-24. 
9. ^ Červený, V. (July 2002). "Fermat's Variational Principle for Anisotropic Inhomogeneous Media". Studia Geophysica et Geodaetica 46 (3): 567. doi:10.1023/A:1019599204028. http://www.ingentaconnect.com/content/klu/sgeg/2002/00000046/00000003/00450806. 
10. ^ Simmons, George F. (2007). Calculus Gems: Brief Lives and Memorable Mathematics. Mathematical Association of America. p. 98. ISBN 0883855615. 
11. ^ Weil 1984, p.104
12. ^ Weil 1984, p.105

References

Books referenced

• Mahoney, Michael Sean (1994). The mathematical career of Pierre de Fermat, 1601 - 1665. Princeton Univ. Press. ISBN 0691036667. 
• Weil, André (1984). Number Theory: An approach through history From Hammurapi to Legendre. Birkhäuser. ISBN 0817631410. 

Further reading

• Singh, Simon (2002). Fermat's Last Theorem. Fourth Estate Ltd. ISBN 1841157910. 

External links

• O'Connor, John J.; Robertson, Edmund F., "Pierre de Fermat", MacTutor History of Mathematics archive .
• The Life and times of Pierre de Fermat (1601 - 1665) from W. W. Rouse Ball's History of Mathematics
• The Mathematics of Fermat's Last Theorem
• Fermat's Achievements
• Fermat's Fallibility at MathPages
• History of Fermat's Last Theorem (French)

Persondata
NAME	Fermat, Pierre de
ALTERNATIVE NAMES
SHORT DESCRIPTION	French mathematician and lawyer
DATE OF BIRTH	17 August 1601
PLACE OF BIRTH	Beaumont-de-Lomagne, France
DATE OF DEATH	12 January 1665
PLACE OF DEATH	Castres, France

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